AEEE flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

Quadratic formula

For ax²+bx+c=0 with a≠0, the roots are x=[−b±√(b²−4ac)]/(2a). The discriminant b²−4ac decides whether roots are real and distinct, repeated, or complex conjugates.

Sum and product of roots

For ax²+bx+c=0, the sum of roots equals −b/a and the product equals c/a. These Vieta relations let you build a quadratic from its roots and check solutions quickly.

Derivative of sin x

The derivative of sin x with respect to x is cos x, and the derivative of cos x is −sin x. Apply the chain rule when the angle is a function of x.

Integration by parts

The rule ∫u dv = uv − ∫v du transforms a product integral. Choose u using the LIATE order: logarithmic, inverse trigonometric, algebraic, trigonometric, exponential.

Circle centre and radius

The circle x²+y²+2gx+2fy+c=0 has centre (−g,−f) and radius √(g²+f²−c). The radius is real only when g²+f²−c is non-negative.

Conditional probability formula

P(A|B)=P(A∩B)/P(B) for P(B)>0. Rearranged, it gives the multiplication rule P(A∩B)=P(A|B)P(B), which underlies Bayes' theorem and tree diagrams.

Geometric progression infinite sum

An infinite geometric series with first term a and ratio r sums to a/(1−r) only when |r|<1. If |r|≥1 the series diverges and no finite sum exists.

Projectile range formula

A projectile launched at speed u and angle θ has range u²sin2θ/g on level ground. Range is maximum at 45°, and complementary angles give equal ranges.

Work-energy theorem

The net work done on a body equals its change in kinetic energy. For conservative forces this connects to potential energy through F=−dU/dx and conserves mechanical energy.

Parallel resistance rule

Parallel resistors satisfy 1/R_total = 1/R₁ + 1/R₂ + …, so the equivalent resistance is smaller than the smallest branch. The voltage across each parallel branch is the same.

Photoelectric threshold frequency

Emission occurs only when light frequency exceeds ν₀=φ/h, where φ is the work function. Above threshold, maximum kinetic energy is hν−φ and intensity affects current, not energy.

Kirchhoff's loop rule

Around any closed circuit loop, the algebraic sum of potential differences is zero. This expresses energy conservation and, with the junction rule, solves multi-loop circuits.

Lens formula

For a thin lens, 1/f = 1/v − 1/u, where f is focal length and u, v are object and image distances. Correct signs for real and virtual cases must be applied consistently.

Mole and Avogadro number

One mole contains 6.022×10²³ particles. Dividing a sample's mass by its molar mass in g/mol gives moles, which convert to particle number or gas volume.

Ideal gas equation

PV=nRT, with R=8.314 J/(mol·K) and temperature in kelvin. For a fixed amount of gas, P₁V₁/T₁=P₂V₂/T₂ combines Boyle's and Charles's laws.

pH definition

pH=−log[H⁺] for dilute aqueous solutions. At 25 °C, pH+pOH=14, so a solution of pH 3 has [H⁺]=10⁻³ M and pOH 11.

Henderson-Hasselbalch equation

Buffer pH equals pKa + log([salt]/[acid]). A buffer resists pH change on adding small amounts of strong acid or base, and works best when pH is near pKa.

Le Chatelier's principle

A system at equilibrium shifts to oppose a disturbance. Adding reactant or removing product shifts forward; compressing a gas equilibrium favors the side with fewer gas moles; catalysts do not shift it.

Hybridization and steric number

Steric number 2 gives sp (linear), 3 gives sp² (trigonal planar), and 4 gives sp³ (tetrahedral). VSEPR then predicts molecular shape from electron-pair repulsion.

Bond order from MO theory

Bond order equals (bonding electrons − antibonding electrons)/2. Higher bond order means a shorter, stronger bond; bond order zero means the molecule is not stable.

Simple vs compound interest

Simple interest is PRT/100 on the original principal. Compound amount is P(1+R/100)ⁿ; over two years compound interest exceeds simple interest by P(R/100)².

Average speed for equal distances

Travelling two equal distances at speeds x and y gives average speed 2xy/(x+y), the harmonic mean. It is always less than the arithmetic mean (x+y)/2.

Combined work formula

If A alone takes a days and B alone takes b days, working together they finish in ab/(a+b) days. The reasoning is that their rates 1/a and 1/b add.

Successive percentage change

Two successive changes of a% and b% produce a net change of a+b+ab/100 percent. The extra term ab/100 arises because the second change applies to the already changed value.

Alligation rule

When mixing two components at prices cheaper and dearer to obtain a mean price, the quantity ratio is (mean−cheaper):(dearer−mean). It also applies to concentrations and averages.

Combinations versus permutations

nCr=n!/[r!(n−r)!] counts selections where order does not matter. nPr=n!/(n−r)! counts ordered arrangements; permutations of a word with repeated letters divide by factorials of repeats.

Downstream and upstream speed

Downstream speed equals boat speed plus stream speed, upstream equals boat minus stream. Stream speed is half the difference, and boat speed is half the sum.

Subject-verb agreement rule

A verb agrees with its subject in number even when a prepositional phrase intervenes, as in "The box of pens is heavy." Indefinite pronouns like everyone and nobody take singular verbs.

Active to passive conversion

Make the object the subject, change the verb to a form of be plus past participle, and add the agent with "by". The original tense is preserved throughout.

Third conditional structure

The third conditional uses if + had + past participle in the condition and would have + past participle in the result. It describes unreal situations in the past.

Affect versus effect

Affect is normally a verb meaning to influence, while effect is normally a noun meaning result. Confusing them is a common error in formal writing and editing tasks.

Reported speech tense shift

After a past reporting verb, shift the reported tense one step back and change time and place words: now becomes then, today becomes that day, here becomes there.